Lawyers for the women said Rubin and several cohorts worked together to entice women "to travel across the United States to a secret penthouse Rubin controlled in Midtown Manhattan."
"In the penthouse, Rubin would beat and rape the women," the lawyers said.
Rubin's lawyer, Schnur, said the women's attorneys asked Rubin to pay the women $32 million before the complaint was filed.
Schnur, a former prosecutor in the Brooklyn District Attorney's Office, is named as a defendant in the lawsuit.
The suit claims Schnur drafted the nondisclosure agreements "that are provided to the victims prior to meeting Rubin" in an attempt to prevent the alleged crimes committed against the women from being exposed.
Schnur, in a statement, said, "The allegations against Mr. Rubin and his attorney, Ms. Schnur, contained in the complaint are fabricated, baseless and without merit."
"The plaintiff's complaint is a web of lies intended to extort Mr. Rubin," Schnur said. "All appropriate law enforcement agencies, including the FBI, have been contacted."
The FBI's field office in New York did not respond to a request for comment.
Balestriere, the women's lawyer, said, "If she's contacted the FBI, I'll speak to them any day of the week."
In another layer of claims, Schnur said that on Aug. 11, a Maryland man named Robert Aloi emailed a board member of a charitable organization that Rubin is involved in.
Aloi, whom Schnur described as a friend and former employer of two of the plaintiffs, told the board member that it was urgent that Rubin speak with Aloi about one of the plaintiffs, Schnur said.
Schnur says she spoke with Aloi via phone on Aug. 14. A spokesman for Schnur said Aloi asked for $9.95 million in exchange for keeping quiet about Rubin's alleged activities with the women.
On Aug. 21, Schnur said, she contacted law enforcement about Aloi's alleged extortion effort against Rubin. Schnur said that several weeks later, on Sept. 15, Aloi canceled a meeting with her that had been scheduled for Sept. 18.
On Sept. 18, a warrant was signed for Aloi's arrest on a charge of theft by extortion by a judge in Middlesex County, New Jersey.
That warrant, provided to CNBC by the Middlesex County prosecutor's office, said that Aloi, between Aug. 11 and Sept. 18, engaged in extortionate conduct by demanding in excess of $75,000 "from the victim in return for not exposing any secret or asserted fact, whether true or false."
The victim was not identified by name in the warrant.
A spokeswoman for the prosecutor's office said she could not comment on whether Rubin was the victim in Aloi's case.
The spokeswoman also said Aloi is due to surrender Monday in New Jersey and is slated to appear in court the following day.
When CNBC called Aloi's home in Maryland, a woman who answered said she would get him to the phone. After a minute or so, the woman returned to the line and said Aloi was "no longer available to talk."
When told that a reporter wanted to speak to Aloi about allegations that he tried to extort money from Rubin, the woman said "OK," and took the reporter's contact information.
Aloi did not return that request for comment, nor did he respond to an email sent to his company.
Schnur said that the women's lawyer, Balestriere, on Sept. 19, emailed to her a draft of the civil lawsuit that ended up being filed by him last week in Brooklyn federal court.
For his part, Balestriere said, "our clients have nothing to do with him," when asked about Aloi. He said that any contact Aloi may have had with Rubin or Schnur was done without the participation of his clients or their knowledge.
Asked why his clients had returned to New York to see Rubin even after experiencing violence at his hands, Balestriere said the women "were very afraid of this man." The lawyer said that even before the women met Rubin, he had obtained "a lot of information about them," including their dates of birth and addresses.
"They were afraid of not cooperating" with Rubin, and also believed, erroneously, that nondisclosure agreements they had signed prevented them from telling lawyers what was happening to them, Balestriere said.